example of cohomology and Mayer-Vietoris sequence

Consider n-dimensional sphere $S^{n}=\{v\in R^{n+1}|\quad|v|=1\}$.

Let $A=\{(x_{0},...,x_{n})\in S^{n}|\quad x_{0}>-1/2\}$ and $B=\{(x_{0},...,x_{n})\in S^{n}|\quad x_{0}<1/2\}$.

Of course both $A$ and $B$ are open (in $S^{n}$) and their union is $S^{n}$. Furthermore, it can be easily seen, that their intersection can be contracted into ”big circle”, i.e. $A\cap B$ has homotopy type of $S^{n-1}$. Also both $A$ and $B$ are contractible (they are homeomorphic to $R^{n}$ via stereographic projection). So, write part of a Meyer-Vietoris sequence (for the cohomology $H^{m}(X)=H^{m}(X,G)$, where $G$ is a fixed Abelian group):

$\cdots\rightarrow H^{m}(A)\oplus H^{m}(B)\rightarrow H^{m}(A\cap B)\rightarrow H% ^{m+1}(S^{n})\rightarrow H^{m+1}(A)\oplus H^{m+1}(B)\rightarrow\cdots$

Since both $A$ and $B$ are contractible and $A\cap B$ is homotopic to $S^{n-1}$, we have the following short exact sequence:

$0\rightarrow H^{m}(S^{n-1})\rightarrow H^{m+1}(S^{n})\rightarrow 0$

which shows that $H^{m}(S^{n-1})$ is isomorphic to $H^{m}(S^{n})$ for every $n>0$ and $m>0$. So, in order to calculate cohomology groups of spheres, we only need to know the cohomology groups of $S^{1}$. And those can be also calculated, if we once again apply previous schema. Note, that in the case of $S^{1}$ we have that $A\cap B$ has the homotopy type of a discrete space with two points. Therefore all their cohomology groups are trivial, except for $H^{0}$ (which can be easily calculated to be equal to $H^{0}(*)\oplus H^{0}(*)$, where * is a one-pointed space).

This schema can be used for other spaces like the torus (which can be also calculated from Kunneth’s formula).

Title example of cohomology and Mayer-Vietoris sequence ExampleOfCohomologyAndMayerVietorisSequence 2013-03-22 19:13:27 2013-03-22 19:13:27 joking (16130) joking (16130) 7 joking (16130) Example msc 55N33